By the Cauchy–Schwarz inequality and the Plancherel theorem
\( |\widetilde{m}_{{\varepsilon }, j}(D)f(x)| \lesssim \Vert \widetilde{m}_{{\varepsilon }, j}\Vert _{L^2(\mathbb {R}^d)} \Vert  \widehat{f}\Vert _{L^2(\mathbb {R}^d)} \lesssim (2^j{\varepsilon })^{\frac{1}{2}-k} \Vert f\Vert _{L^2({\mathbb {R}}^d)}. \) 
Hence, we have the estimate (REF ).
Let us write \(\psi _j(t)=\psi (2^{-j}t)\)  for \(j\ge 1\) . Setting \(x=(y,t)\)  and using the spherical coordinates we note that \(\widetilde{m}_{{\varepsilon }, j}(D)f(x)\)  is equal to
\( \frac{1}{(2\pi )^d} \int _0^\infty \psi _j \bigg (\frac{\rho ^2-1}{{\varepsilon }}\bigg ) \int _{\mathbb {R}}\int _{S^{d-2}} e^{i(\rho y\cdot \theta +t\tau )} \big (\widetilde{m}_{{\varepsilon }}\widehat{f} \,\big )(\rho \theta , \tau ) d\sigma (\theta ) d\tau \rho ^{d-2}d\rho . \) 
Thus, by Minkowski's inequality \(\Vert \widetilde{m}_{{\varepsilon }, j}(D)f\Vert _{L^\frac{2d}{d-2}({\mathbb {R}}^d)}\)  is bounded by a constant times
\(\int _0^\infty \psi _j \bigg (\frac{\rho ^2-1}{{\varepsilon }}\bigg ) \left\Vert  \int _{\mathbb {R}}\int _{S^{d-2}} e^{i(\rho y\cdot \theta +t\tau )} \big (\widetilde{m}_{{\varepsilon }}\widehat{f} \,\big )(\rho \theta , \tau ) d\sigma (\theta ) d\tau \right\Vert _{L_{y,t}^\frac{2d}{d-2}} \rho ^{d-2}d\rho .\) 
Applying Hausdorff–Young's and Minkowski's inequalities successively gives
\(&\left\Vert  \int _{\mathbb {R}}\int _{S^{d-2}} e^{i(\rho y\cdot \theta +t\tau )} \big (\widetilde{m}_{{\varepsilon }}\widehat{f} \,\big )(\rho \theta , \tau ) d\sigma (\theta ) d\tau \right\Vert _{L_{y,t}^{\frac{2d}{d-2}}({\mathbb {R}}^d)} \\&\lesssim \bigg \Vert  \left\Vert  \int _{S^{d-2}} e^{i\rho y\cdot \theta } \big (\widetilde{m}_{{\varepsilon }}\widehat{f} \,\big )(\rho \theta , \tau ) d\sigma (\theta ) \right\Vert _{L_\tau ^\frac{2d}{d+2}({\mathbb {R}})} \bigg \Vert _{L_{y}^{\frac{2d}{d-2}}({\mathbb {R}}^{d-1})} \\&\le \bigg \Vert  \left\Vert  \int _{S^{d-2}} e^{i\rho y\cdot \theta } \big (\widetilde{m}_{{\varepsilon }}\widehat{f} \,\big )(\rho \theta , \tau ) d\sigma (\theta ) \right\Vert _{L_{y}^{\frac{2d}{d-2}}({\mathbb {R}}^{d-1})} \bigg \Vert _{L_\tau ^\frac{2d}{d+2}({\mathbb {R}})} .\) 

Making use of the adjoint restriction estimate due to Tomas [1]} and Stein [2]} and the Hölder inequality, the last is
\( \lesssim \big \Vert  \big \Vert \big (\widetilde{m}_{{\varepsilon }}\widehat{f} \,\big )(\rho \theta , \tau )\big \Vert _{L^2_\theta (S^{d-2})} \big \Vert _{L_\tau ^\frac{2d}{d+2}({\mathbb {R}})} \lesssim \big \Vert \big (\widetilde{m}_{{\varepsilon }}\widehat{f} \,\big )(\rho \theta , \tau )\big \Vert _{L^2_{\theta , \tau }(S^{d-2}\times {\mathbb {R}})}. \) 
Since the \(\rho \) -support of the integrand in (REF ) is contained in an interval of length \(\sim 2^j{\varepsilon }\)  and \(\Vert \widetilde{m}_{{\varepsilon },j}\Vert _{L^\infty }\lesssim (2^j{\varepsilon })^{-k} \) , the Cauchy–Schwarz inequality and the Plancherel theorem yield
\(\Vert \widetilde{m}_{{\varepsilon }, j}(D)f\Vert _{L^\frac{2d}{d-2}}&\lesssim \sqrt{2^j{\varepsilon }} \bigg (\int _0^\infty \psi _j \bigg (\frac{\rho ^2-1}{{\varepsilon }}\bigg )^2 \big \Vert \big (\widetilde{m}_{{\varepsilon }}\widehat{f} \,\big )(\rho \theta , \tau )\big \Vert _{L^2_{\theta , \tau }(S^{d-2}\times {\mathbb {R}})}^2 \rho ^{d-2}d\rho \bigg )^\frac{1}{2} \\&\le \sqrt{2^j{\varepsilon }} \, \big \Vert  \big ( \widetilde{m}_{{\varepsilon },j} \widehat{f} \,\big )(\eta , \tau )\big \Vert _{L^2_{\eta , \tau }(\mathbb {R}^d)} \lesssim (2^j{\varepsilon })^{\frac{1}{2}-k} \Vert f\Vert _{L^2}.\) 

This gives the estimate ().
Finally, we prove the estimate (REF ). Using the spherical coordinates as before we have
\( \widetilde{m}_{{\varepsilon }, j}(D)f(y,t)= \frac{1}{(2\pi )^d} \int _0^\infty \int _{\mathbb {R}}e^{it\tau }A_{{\varepsilon }, j}(\rho , \tau ) \int _{S^{d-2}} e^{i\rho \theta \cdot y} \widehat{f}(\rho \theta , \tau ) d\sigma (\theta ) d\tau d\rho , \) 
where
\( A_{{\varepsilon },j}(\rho , \tau ) = \psi _j\bigg (\frac{\rho ^2-1}{{\varepsilon }}\bigg ) \frac{\rho ^{d-2} \psi _0({\varepsilon }_\circ ^{-1}(\rho ^2-1))\psi (\tau )}{(\rho ^2-1+{\varepsilon }^2\tau ^2+2i{\varepsilon }\tau )^k}. \) 
Since \(\psi _j(t)\psi _0(t/2^{j+1})=\psi _j(t)\) , we may write
\( \widetilde{m}_{{\varepsilon }, j}(D)f(y,t)= \frac{1}{(2\pi )^d} \int _{I_j} \int _{\mathbb {R}}e^{it\tau }A_{{\varepsilon }, j}(\rho , \tau ) \int _{S^{d-2}} h_j(\rho \theta ; y,\tau ) d\sigma (\theta ) d\tau d\rho , \) 
where
\( h_j(\eta ;y,\tau ):=e^{i\eta \cdot y} \widehat{f}(\eta , \tau )\psi _0 \bigg (\frac{|\eta |^2-1}{2^{j+1}{\varepsilon }}\bigg ) \) 
and
\( I_j :={\left\lbrace \begin{array}{ll}\big [\sqrt{1-2^{j+1}{\varepsilon }}\,,\sqrt{1-2^{j-1}{\varepsilon }} \,\big ] \cup \big [ \sqrt{1+2^{j-1}{\varepsilon }}\, ,\sqrt{1+2^{j+1}{\varepsilon }} \,\big ] &\text{if} \ \ j\ge 1, \\[3pt]\big [\sqrt{1-2{\varepsilon }}\,,\sqrt{1+ 2{\varepsilon }} \,\big ] &\text{if} \ \ j=0.\end{array}\right.} \) 
From the identities (REF ) and (REF ) (with \(n=d-1\) ) it follows that
\(\widetilde{m}_{{\varepsilon }, j}(D)f(y,t)&= c_d \int _{\mathbb {R}}\int _{I_j} e^{it\tau }A_{{\varepsilon }, j}(\rho , \tau ) \big \langle \lambda ^\ast \chi _+^{-k}, L^{-(k-1)} h_j(\rho \theta ; y, \tau ) \big \rangle _\theta d\rho d\tau \) 

for some constant \(c_d\)  depending only on \(d\) . Here we are slightly abusing notations. In fact,
\( L^{-(k-1)} h_j(\rho \theta ; y, \tau )=2^{k-1}\int _{[0,1]^{k-1}} (s_1\cdots s_{k-1})^{d-4} h_j( \rho s_1\cdots s_{k-1} \theta ; y, \tau )d\mathbf {s}, \) 
where \(d\mathbf {s} = ds_1\cdots ds_{k-1}\)  (see (REF )).
The domain of integration \([0,1]^{k-1}\)  can be replaced by \([1-2^{j+2}{\varepsilon },1]^{k-1}\) . To see this, we note \(\mathrm {supp\,}\lambda ^* \chi _+^{-k} \subset S^{d-2}\) , so \(h_j (\rho s_1\cdots s_{k-1}\theta ; y, \tau )\ne 0\)  only if
\( \rho ^{-1} \sqrt{1-2^{j+2}{\varepsilon }} \le s_1\cdots s_{k-1} \le \rho ^{-1} \sqrt{1+2^{j+2}{\varepsilon }}. \) 
Since \(\rho \in I_j\) , we have \(s_l\ge 1-2^{j+2}{\varepsilon }\)  for all \(l=1,\ldots , k-1\) .
Therefore,
\(&|\widetilde{m}_{{\varepsilon }, j}(D)f(y,t)| \\&\lesssim \int _{[1-2^{j+2}{\varepsilon },1]^{k-1}} \int _{I_j} \bigg | \int _{\mathbb {R}}e^{it\tau }A_{{\varepsilon }, j}(\rho , \tau ) \big \langle \lambda ^\ast \chi _+^{-k}, h_j(\rho s_1\cdots s_{k-1}\theta ;y,\tau )\big \rangle _\theta d\tau \bigg | d\rho d\mathbf {s} \\&\le \int _{[1-2^{j+2}{\varepsilon },1]^{k-1}} \int _{I_j} \bigg | \int _{\mathbb {R}}e^{it\tau }A_{{\varepsilon }, j}(\rho , \tau ) \big \langle \lambda ^\ast \chi _+^{-k}, \widetilde{h}(\rho s_1\cdots s_{k-1}\theta ;y,\tau )\big \rangle _\theta d\tau \bigg | d\rho d\mathbf {s} ,\) 

where
\(\widetilde{h}(\eta ;y,\tau )=\widehat{f}(\eta , \tau )e^{i\eta \cdot y}.\) 
Making use of Lemma REF , we obtain
\( \Vert \widetilde{m}_{{\varepsilon },j}(D)f\Vert _{L^{q,\infty }({\mathbb {R}}^d)} \lesssim (2^j{\varepsilon })^{k} \sup _{\rho \in I_j}\Vert A_{{\varepsilon },j}(\rho ,\cdot )\Vert _{C^2} \Vert f\Vert _{L^{p,1}({\mathbb {R}}^d)} \) 
for \((\frac{1}{p},\frac{1}{q})\in [B_{k}^d, D_{k}^d]\) . It is easy to check that \(\sup _{\rho \sim 1} \Vert A_{{\varepsilon },j}(\rho ,\cdot )\Vert _{C^2}\lesssim (2^j{\varepsilon })^{-k}\) . Therefore (REF ) follows.
